We can interpret a 3-dimensional handlebody \(C_k\)  as a cobordism from a closed surface \(F_k\)  to the empty set. This way, Definition REF  generalizes Definition REF .
For a detailed discussion on the properties of relative trisections one can see [1]}, [2]}, [3]}.
The authors of this paper like to think of trisections of 4-manifolds as decompositions into three `standard' pieces \(Z_i\)  glued along submanifolds of their boundary.
In this paper we will see some interactions between relative and \(\star \) -trisections.
In the following subsection we will describe the standard models for the 4-manifold pieces in a \(\star \) -trisection.
